Estimating population values ppt @ bec doms

Chapter Goals ,[object Object],[object Object],[object Object],[object Object],[object Object]

Confidence Intervals ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Point and Interval Estimates ,[object Object],[object Object],Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval

Point Estimates We can estimate a Population Parameter … with a Sample Statistic (a Point Estimate) Mean Proportion p p x μ

Confidence Intervals ,[object Object],[object Object],[object Object]

Confidence Interval Estimate ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Estimation Process (mean, μ, is unknown) Population Random Sample Mean x = 50 Sample I am 95% confident that μ is between 40 & 60.

General Formula ,[object Object],Point Estimate  (Critical Value)(Standard Error)

Confidence Level ,[object Object],[object Object],[object Object]

Confidence Level, (1-  ) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],(continued)

Confidence Intervals Population Mean σ Unknown Confidence Intervals Population Proportion σ Known

Confidence Interval for μ ( σ Known) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Finding the Critical Value ,[object Object],z . 025 = -1.96 z . 025 = 1.96 Point Estimate Lower Confidence Limit Upper Confidence Limit z units: x units: Point Estimate 0

Common Levels of Confidence ,[object Object],Confidence Level Confidence Coefficient, z value, 1.28 1.645 1.96 2.33 2.57 3.08 3.27 .80 .90 .95 .98 .99 .998 .999 80% 90% 95% 98% 99% 99.8% 99.9%

Interval and Level of Confidence Confidence Intervals Intervals extend from to 100(1-  )% of intervals constructed contain μ ; 100  % do not. Sampling Distribution of the Mean x x 1 x 2

Margin of Error ,[object Object],Example: Margin of error for estimating μ, σ known:

Factors Affecting Margin of Error ,[object Object],[object Object],[object Object]

Example ,[object Object],[object Object]

Example ,[object Object],[object Object],(continued)

Interpretation ,[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],Confidence Interval for μ ( σ Unknown)

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Confidence Interval for μ ( σ Unknown) (continued)

Student’s t Distribution ,[object Object],[object Object],[object Object],[object Object]

Degrees of Freedom (df) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],If the mean of these three values is 8.0, then x 3 must be 9 (i.e., x 3 is not free to vary) Here, n = 3, so degrees of freedom = n -1 = 3 – 1 = 2 (2 values can be any numbers, but the third is not free to vary for a given mean)

Student’s t Distribution t 0 t ( df = 5) t ( df = 13) t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal Standard Normal (t with df =  ) Note: t z as n increases

Student’s t Table Upper Tail Area df .25 .10 .05 1 1.000 3.078 6.314 2 0.817 1.886 2.920 3 0.765 1.638 2.353 t 0 2.920 The body of the table contains t values, not probabilities Let: n = 3 df = n - 1 = 2  = .10  /2 =.05  /2 = .05

t distribution values With comparison to the z value Confidence t t t z Level (10 d.f.) (20 d.f.) (30 d.f.) ____ .80 1.372 1.325 1.310 1.28 .90 1.812 1.725 1.697 1.64 .95 2.228 2.086 2.042 1.96 .99 3.169 2.845 2.750 2.57 Note: t z as n increases

Example ,[object Object],[object Object],[object Object],[object Object],46.698 …………….. 53.302

Approximation for Large Samples ,[object Object],Technically correct Approximation for large n

Determining Sample Size ,[object Object],[object Object],[object Object]

Required Sample Size Example ,[object Object],(Always round up) So the required sample size is n = 220

If σ is unknown ,[object Object],[object Object],[object Object]

Confidence Intervals for the Population Proportion, p ,[object Object]

Confidence Intervals for the Population Proportion, p ,[object Object],[object Object],(continued)

Confidence interval endpoints ,[object Object],[object Object],[object Object],[object Object],[object Object]

Example ,[object Object],1. 2. 3. (continued)

Interpretation ,[object Object],[object Object],[object Object]

Changing the sample size ,[object Object],[object Object],[object Object],[object Object]

Finding the Required Sample Size for proportion problems Solve for n: Define the margin of error: p can be estimated with a pilot sample, if necessary (or conservatively use p = .50)

What sample size...? ,[object Object],[object Object]

What sample size...? Solution: For 95% confidence, use Z = 1.96 E = .03 p = .12 , so use this to estimate p So use n = 451 (continued)

Using PHStat ,[object Object],[object Object],[object Object]

PHStat Interval Options options

Using PHStat (for μ, σ unknown) A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ

Using PHStat (sample size for proportion) How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (Assume a pilot sample yields p = .12)

Estimating population values ppt @ bec doms

Recomendados

Recomendados

Mais conteúdo relacionado

Mais procurados

Mais procurados (20)

Destaque

Destaque (12)

Semelhante a Estimating population values ppt @ bec doms

Semelhante a Estimating population values ppt @ bec doms (20)

Mais de Babasab Patil

Mais de Babasab Patil (20)

Último

Último (20)

Estimating population values ppt @ bec doms